rocal curves Flashcards","topicId":"3f3e4fbe-a331-4a71-a9cd-e076d16c6b9f","topicSlug":"sketching-reciprocal-curves","cards":[{"id":"7ea951ab-4f0a-4dd9-a606-3a851f0352df","cardType":"basic","front":"Why is $x=0$ excluded from the domain of $y=\\frac{a}{x}$ and $y=\\frac{a}{x^2}$?","back":"It would cause division by zero, which is undefined.","points":2,"mcqOptions":null},{"id":"f1c7d1d6-bd47-41e4-860f-bf600f76309a","cardType":"basic","front":"Why does each reciprocal curve have two separate branches?","back":"The excluded input $x=0$ separates negative and positive $x$-values.","points":2,"mcqOptions":null},{"id":"75d17b9e-fc82-4fb4-8c2f-bcdb587c2f76","cardType":"basic","front":"What is the vertical asymptote of both reciprocal families?","back":"$x=0$","points":1,"mcqOptions":null},{"id":"6295d5e8-91fc-4273-a446-7ef1d0764e26","cardType":"basic","front":"What is the horizontal asymptote of both reciprocal families?","back":"$y=0$","points":1,"mcqOptions":null},{"id":"f77fecfa-38f8-4cba-9446-dc08a2dc7719","cardType":"basic","front":"Why does a reciprocal curve with $a\\neq0$ cross neither coordinate axis?","back":"

$x=0$ is excluded.
$y=0$ is impossible for non-zero $a$.

","points":2,"mcqOptions":null},{"id":"33a7ff0f-ef1d-4ccf-a903-0ed633f2c2c7","cardType":"basic","front":"For $y=\\frac{a}{x}$ with $a>0$, which quadrants contain the branches?","back":"Quadrants I and III","points":1,"mcqOptions":null},{"id":"c92dca0e-e573-42ee-84df-14d9afe81f43","cardType":"basic","front":"For $y=\\frac{a}{x}$ with $a<0$, which quadrants contain the branches?","back":"Quadrants II and IV","points":1,"mcqOptions":null},{"id":"e3da208c-8ec2-4ac7-a9c0-88f93903bd29","cardType":"basic","front":"What symmetry does $y=\\frac{a}{x}$ have?","back":"Rotational symmetry of order two about the origin.","points":1,"mcqOptions":null},{"id":"30f4402d-f9ff-405d-8c86-efe19cbaf325","cardType":"basic","front":"Why is the sign of $y=\\frac{a}{x^2}$ controlled entirely by $a$?","back":"For every permitted $x$, $x^2>0$.","points":2,"mcqOptions":null},{"id":"d9967cc5-79be-4b49-a726-9ae87b0c7c78","cardType":"basic","front":"For $y=\\frac{a}{x^2}$ with $a>0$, which quadrants contain the branches?","back":"Quadrants I and II","points":1,"mcqOptions":null},{"id":"c228d851-28ca-4db1-8fd0-48e32834e954","cardType":"basic","front":"For $y=\\frac{a}{x^2}$ with $a<0$, which quadrants contain the branches?","back":"Quadrants III and IV","points":1,"mcqOptions":null},{"id":"4ce574cb-9c8f-4727-a95b-01fa8e39b61f","cardType":"basic","front":"What symmetry does $y=\\frac{a}{x^2}$ have?","back":"Reflection symmetry in the $y$-axis.","points":1,"mcqOptions":null},{"id":"b7f9907c-dee9-41b6-9e8a-bf4883409235","cardType":"basic","front":"For fixed non-zero $x$, what happens when $|a|$ increases?","back":"$|y|$ increases, moving the point further from the $x$-axis.","points":2,"mcqOptions":null},{"id":"8d395809-5fdf-4b40-a0b9-1a7dc998d496","cardType":"basic","front":"Does changing $|a|$ change the reciprocal curve's asymptotes?","back":"No - they remain $x=0$ and $y=0$.","points":1,"mcqOptions":null},{"id":"61356f42-6126-40a9-b2e7-142bdd61a679","cardType":"cloze","front":"As $x$ approaches zero, the denominator becomes very small, so [ ].","back":"As $x$ approaches zero, the denominator becomes very small, so $|y|$ becomes very large.","points":null,"mcqOptions":null},{"id":"3bd11940-e9b5-4672-b469-a8321afbbc8d","cardType":"cloze","front":"As $|x|$ becomes very large, a reciprocal curve approaches $\\text{[...]}$.","back":"As $|x|$ becomes very large, a reciprocal curve approaches $\\boxed{y=0}$.","points":null,"mcqOptions":null},{"id

1.2.15 Sketching reciprocal curves Flashcards

1.2.15 Sketching reciprocal curves Flashcards